Unit RATIONAL MECHANICS I
- Course
- Mathematics
- Study-unit Code
- 55031206
- Curriculum
- In all curricula
- Teacher
- Francesca Di Patti
- Teachers
-
- Francesca Di Patti
- Hours
- 63 ore - Francesca Di Patti
- CFU
- 9
- Course Regulation
- Coorte 2020
- Offered
- 2022/23
- Learning activities
- Caratterizzante
- Area
- Formazione modellistico-applicativa
- Academic discipline
- MAT/07
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Kinematics: kinematics of the point, kinematics of material systems and the rigid body, relative kinematics.
Dynamics: principles, statics and dynamics of the point, statics and dynamics of constrained systems.
Dynamics of material systems: geometry of the masses, Euler's equations, first integrals.
Analitica mechanics: D'Alembert's principle, Lagrange equations, Hamilton equations, Stability and small oscillations. Canonical transformations. - Reference texts
- M. FABRIZIO, Elementi di Meccanica Classica, Zanicchelli, 2020.
H. GOLDSTEIN, C.P. POOLE, J.L. SAFKO, Classical Mechanics, III ed., Addison Wesley, 2001.
G. GRIOLI, Lezioni di Meccanica Razionale, Libreria Cortina, 2002.
V. I. ARNOLD, Mathematical Methods of Classical Mechanics, II ed., Springer-Verlag, 1989.
F. R. GANTMACHER, Lezioni di Meccanica Analitica, Editori Riuniti, 1980.
M. BRAUN, Differential Equations and their Applications, IV ed., Springer-Verlag, 1993. - Educational objectives
- The course aims to give the students the fundamental mathematical instruments and methods useful for studying mechanical systems, in particular kinematics, statics and dynamics of a point particle and of the rigid body and Lagrangian and Hamiltonian mechanics. Students will become familiar with the mathematical modelling of mechanical systems and will learn the analytical methods useful to study the dynamics and statics of such systems.
- Prerequisites
- Basic knowledge of euclidean geometry, calculus, algebra and classical mechanics
- Teaching methods
- Lectures on all subjects of the course and respective exercises. Supporting material and a detailed program will be posted on unistudium.
- Other information
- It is recommended to attend the lectures.
- Learning verification modality
- Written and oral exam. The written test consists of 2 exercises of the same level of those solved during the classes. The pass mark for the written exam is 16/30. The oral exam can be postponed to any session within the end of the current academic year. The oral examination consists in an interview about 2/3 theoretical aspects. Operating time up to 30 minutes.
During the course, students can perform two optional intermediate tests. If in both tests they get a final mark greater or equal 16/30, they can skip the written exams and sit the oral exam.
For information about services for students with disabilities and/or DSA visit the page http://www.unipg.it/disabilita-e-dsa. - Extended program
- KINEMATICS
Trajectory, velocity and acceleration. Relative kinematics and angular velocity vector. Addition of angular velocities. Velocity and acceleration fields of a rigid motion. Instantaneous axis of rotation. Planar rigid motions.
DYNAMICS OF THE SYSTEMS OF POINT MASSES AND RIGID BODIES
Kinetic energy, work, potential energy, kinetic energy Theorem and conservation of mechanical energy for systems of point masses. Cardinal equations of dynamics. Angular momentum and kinetic energy of a rigid body. Geometry of the masses: polar, axial and centrifugal moments of inertia. Matrix of inertia and ellipsoid of inertia. Principal axes of inertia. Rigid bodies rotating around a fixed axis.
DYNAMICS OF CONSTRAINED SYSTEMS
Ideal holonomic constraints for a system of point masses. Degrees of freedom, Lagrangian coordinates and tangent space. Principle of virtual works and its geometric interpretation. Symbolic equation of the dynamics and Lagrange equations of second kind. Lagrangian function. Quadratic structure of the kinetic energy. Hamiltonian formulation of the Lagrange equations. Generalized Dirichlet criterion. Quadratic approximation of the Lagrangian around a stable equilibrium and linearized equations of motion. Normal modes and frequency of the small oscillations around the stable equilibrium configuration.
LAGRANGIAN MECHANICS
Space of configurations and generalized coordinates. Holonomic and nonholonomic constraints. The principle of virtual work, and the principle of d'Alembert. The Lagrange equations, with or without non-conservative generalized forces.